Does falsifiability require the process of falsification be finished
in finite time?
Awesome thinking. The best answer I know of is no.
I'll make my case.
Take the complement of your question:
Does truifiability require the process of truification be finished in finite time?
This is equivalent to your question if one simply negates the claim under test. In other words, is it verifiable that there exists a human who lives forever?
More generally, if a claim is true, is it only verifiable if the search space is effectively finite?
Using a different example so as to avoid the explicit requirement of infinite time, take the claim that
a certain element never seen before by mortals exists.
If the universe is infinite in extent, the search for it would never terminate until either (a) it is found or (b) the searchers give up.
If it does not exist, condition (a) will never be met, and the only way the search will end is by giving up the search, having neither proved nor disproved the claim.
If it does exist, condition (a) is a possibility, however, depending on its rarity or the difficulty of verification, condition (b) is also possible, which again would neither prove nor disprove the claim.
The only possible proof in this case is positive proof: At least by the sole means of direct examination of the search space, disproving the existence of such an element would be impossible, hence the claim of its existence is unfalsifiable.
Now it is clear that there are at least two possible realities for an unfalsifiable claim:
- It is either verifiable and therefore true, or
- It is not verifiable (and may be false).
Even though no finite time limit can be given within which a search through an infinite space for a rare element can prove that it exists, at least if it does exist, it may still be proven to exist in finite time, and therefore its existence is verifiable. The same argument applies to any falsification which could take an unbounded amount of time. It may still be falsifiable, although the expense of falsification is unbounded.
How do humans deal with such issues of observational rarity in practice?
Through the very human-sounding attributes of desire, belief, and patience.
It's pretty practical, and pretty apparent that if you really want to find gold, and if you believe you can or will find it, you're much more likely to keep looking for it than someone who either doesn't want it so much, or who doesn't believe he will or even might find it.
It's partly your choice
So, at a personal level, the observational falsifiability of claims within a potentially infinite search space depends on such subjective, intentional properties such as patience, confidence, and belief.
This illustrates a blind spot.
A given claim is either true or it is not true. An unfalsifiable claim could be true, or false. An unverifiable claim could be true, or false.
Engaging in the search for a positive or negative proof is betting one's resources and energy on the outcome which may be uncertain from the start. A person who never gives up the search for a difficult truth of great value would be shown to be one of the wisest of the race. A person who never gives up the search for a proof something that is untrue would be among the most foolish. Hence we have such a sharp division of opinions and even persecution in matters both scientific and religious, and it is truly said that a genius is often not appreciated in his own time, and no prophet is accepted in his own country.
Anyway, you've just proved that subjective attributes such as faith, belief, and patience necessarily take an important place in the process of scientific discovery.
We can't number the life-saving breakthroughs that others thought couldn't be done.
We also might have difficulty enumerating the vast list of false beliefs people hold that will never be proven; for example, abiogenesis.
You've also hit upon the Decidability Problem of computer science. In essence it says that there may be true statements that exist but which cannot be proven in finite time through methods of exhaustive search, because the search space is infinite. One can decide to embark upon the search for something that may or may not exist. If it does exist, although the tests may appear to be infinite, one's faith is eventually going to be rewarded. If it does not exist, the real test of wisdom is how quickly one abandons the search.
Back to the coin
For the example you gave, the best a person can do in finite time is to take sufficient measurements so as to justify a level of confidence (which is a belief) in a degree of fairness, to some finite level of precision. Of course the possibility is nonzero that any such conclusion is entirely wrong, unless a property can be determined in finite time.
The fair coin is actually a problem that to some degree suffers from a problem of being neither falsifiable nor truifiable, that is, through sampling alone one could never determine that it is fair or unfair.
Depending on the degree of precision and confidence (these are subjective choices), one could tentatively "prove" or "disprove" the coin's fairness based upon experience, and even assign a probability to that judgment. With predetermined margins for error, the probability of either fairness or unfairness could be vanishingly small, and so this is a matter for confidence: How confident do you want to be?